A216778 Number of derangements on n elements with an even number of cycles.
1, 0, 0, 0, 3, 20, 130, 924, 7413, 66744, 667476, 7342280, 88107415, 1145396460, 16035550518, 240533257860, 3848532125865, 65425046139824, 1177650830516968, 22375365779822544, 447507315596451051, 9397653627525472260, 206748379805560389930, 4755212735527888968620
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..450
- Paulo H. L. Martins, Ronald Dickman, and Robert M. Ziff, Percolation in two-species antagonistic random sequential adsorption in two dimensions, arXiv:2211.04622 [cond-mat.stat-mech], 2022.
Programs
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Maple
a := proc (n) local x, y, t, k; if n = 0 then 1 elif n = 1 then 0 else x := 1; y := 0; for k from 2 to n do t := y; y := (k-1)*(x+y+k-3); x := t end do; y end if end proc;
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Mathematica
nn=23;Range[0,nn]!*CoefficientList[Series[Cosh[Log[1/(1-x)]-x],{x,0,nn}],x] (* Geoffrey Critzer, Jun 23 2014 *)
Formula
a(n+1) = n*(a(n) + a(n-1) + n - 2), a(0)=1, a(1)=0.
a(n) = (A000166(n) - n + 1)/2.
E.g.f.: cosh(log(1/(1-x)) - x). - Geoffrey Critzer, Jun 23 2014